COMPUTATION OF MAXIMUM-LIKELIHOOD-ESTIMATES IN ASSOCIATION MODELS

In association models for cross-classified data, computation of maximu m likelihood estimates (MLE's) is relatively difficult due to-the nonl inear constraints on the parameters. Currently available procedures ba sed on the scoring algorithm for constrained maximum likelihood are re latively unreliable, and other cyclic procedures are relatively slow a nd do not provide estimated asymptotic standard deviations as by-produ cts of calculations. To facilitate computations, it is noted that in s tandard association models, removal of constraints results in underide ntification of parameters but does not affect the model itself, so tha t the MLE's of cell probabilities and conditional probabilities are un affected. Given this observation, maximum likelihood estimation may be accomplished by unconstrained maximization of an objective function w ith two components: a log-likelihood ratio and a sum of squares repres enting deviations of parameters from their constraints. The objective function is then maximized by using a modification of the Newton-Raphs on algorithm that ensures that successive iterations increase the obje ctive function whenever a local maximum has not been reached. The prop osed algorithm is shown to be reliable and relatively rapid. In additi on, it is shown that the proposed technique may be used to estimate as ymptotic standard deviations of parameter estimates. Use of the algori thm in practice is illustrated through some standard examples of assoc iation models.